Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness

In this paper, we propose an approach to construct a family of two-dimensional compactly supported real-valued quincunx tight framelets {ϕ; Ψ₁, Ψ₂, Ψ₃} in L₂(ℝ²) with symmetry property and arbitrarily high orders of vanishing moments. Such quincunx tight framelets are associated with quincunx tight framelet filter banks {a; b₁, b₂, b₃} having increasing orders of vanishing moments, possessing symmetry property, and enjoying the additional double canonical properties:

b₁ (k₁, k₂) = (−1)^1+k1+k2 a (1 − k₁, −k₂),
∀ k₁, k₂ ∈ Z
b₃ (k₁, k₂) = (−1)^1+k1+k2 b₂ (1 − k₁, −k₂),

Moreover, the supports of all the high-pass filters b₁, b₂, b₃ are no larger than that of the low-pass filter a. For a low-pass filter a which is not a quincunx orthogonal wavelet filter, we show that a quincunx tight framelet filter bank {a; b₁, . . ., b_L } with b₁ taking the above canonical form must have L ≥ 3 highpass filters. Thus, our family of double canonical quincunx tight framelets with symmetry property has the minimum number of generators. Numerical calculation indicates that this family of double canonical quincunx tight framelets with symmetry property can be arbitrarily smooth. Using one-dimensional filters having linear-phase moments, in this paper we also provide a second approach to construct multiple canonical quincunx tight framelets with symmetry property. In particular, the second approach yields a family of 6-multiple canonical real-valued quincunx tight framelets in L₂(ℝ²) and a family of double canonical complex-valued quincunx tight framelets in L₂(ℝ²) such that both of them have symmetry property and arbitrarily increasing orders of smoothness and vanishing moments. Several examples are provided to illustrate our general construction and theoretical results on canonical quincunx tight framelets in L₂(ℝ²) with symmetry property, high vanishing moments, and smoothness. Quincunx tight framelets with symmetry property constructed by both approaches in this paper are of particular interest for their applications in computer graphics and image processing due to their polynomial preserving property, full symmetry property, short support, and high smoothness and vanishing moments.